The Number of Unlabeled Orders on Fourteen Elements

Lacking an explicit formula for the numbers T0(n) of all order relations (equivalently: T0 topologies) on n elements, those numbers have been explored only up to n = 13 (unlabeled posets) and n = 15 (labeled posets), respectively. In a new approach, we used an orderly algorithm to (i) generate each unlabeled poset on up to 14 elements and (ii) collect enough information about the posets on 13 elements to be able to compute the number of labeled posets on 16 elements by means of a formula by Erné. Unlike other methods, our algorithm avoids isomorphism tests and can therefore be parallelized quite easily. The underlying principle of successively adding new elements to small objects is applicable to lattices and other kinds of order structures, too.